The Intuition vs. Formalism of Mathesis
Doug
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In discussing the trouble with physics, we always come down to the fundamental problem: We don’t understand how nature is both continuous and discrete at the same time. Our modern physical theories have to be either discrete, like quantum mechanics, or continuous, like general relativity. Some theorists, like string theorists, seek to make the discrete continuous, while others, like the loop theorists, seek to make the continuous discrete. Still others, like the particle theorists, seek to keep on keeping on, in search of the ultimate discrete answer, the illusive “God particle.”
It’s an amazing thing to contemplate the billions of dollars mankind is willing to spend to fund the efforts of these relatively small groups of advanced thinkers, in the hope that one of them will figure out how nature does it. What a sight it makes in our minds!
However, in the 19th Century, there was a tiny group of mathematicians, contemplating these things as well. Although, Sir William Rowan Hamilton, Hermann Gunther Grassmann, and William Kingdom Clifford didn’t think of themselves as a group, much less as a group working to solve the mystery of the continuous/discrete duality of nature. Such a dichotomy in nature wasn’t even suspected in their day, but for them the quest to unify the numbers of algebra and the magnitudes of geometry amounted to the same daunting challenge.
For Grassmann, a German high school teacher, the vision was one of a universal geometric calculus, which has finally been realized, in modern times, primarily through Hestenes’ recognition of Clifford’s contribution, wherein he (Clifford) brought the ideas of Hamilton’s quaternions and Grassmann ‘s exterior algebra together in what eventually emerged as Clifford algebras. Hestenes’ vision, he realized in retrospect, was the same as Grassmann’s: “a universal geometric calculus to serve as a unifed language for mathematics and physics.” This “grand vision for mechanics,” asserts Hestenes, “has recently been utterly fulfilled with a complete reformulation of the subject in terms of geometric algebra.” He writes in a paper entitled “Grassmann’s Vision,” that
…geometric calculus embraces a greater range of mathematics than any other mathematical system, including linear and multilinear algebra, projective geometry, distance geometry, calculus on manifolds, hypercomplex function theory, differential geometry, Lie groups and Lie algebras. Geometric calculus has also fulfilled Grassmann’s vision of a universal language for physics. Besides integrating the mathematical formulations of mechanics, relativity and electrodynamics, it has revealed a geometric basis for complex numbers in quantum mechanics.
Yet, this “utter” fulfillment of the “grand vision” of a “unified language for mathematics and physics” clearly has not had much impact on the fundamental problem of the theoretical physicists: The challenge of explaining the unification of the discrete and continuous aspects of nature remains unanswered. Maybe it’s too early to make this judgement. It takes many years for a new development like this to take its place in the tool box of physicists. Yet, even if it does become more widely accepted, it seems clear that simplifying and clarifying the formalisms of quantum mechanics and general relativity doesn’t make the two theories compatible with one another.
They say that the definition of insanity is doing the same thing over and over again, but expecting different results. Yet, that’s what some in the world of theoretical physicists seem to be doing. They seek to find the solution to the crisis in theoretical physics, by reformulating the current theories, over and over again. One of the latest in these attempts was discussed recently on Peter Woit’s blog, Not Even Wrong. It is Chris Isham’s effort to reformulate quantum mechanics, using something called “topos theory.”
In an article entitled “Topos or Not Topos,” Govert Schilling explains that, according to Isham, topos theory “is not a quantum gravity theory itself, but a set of tools to build new theories. It’s a deep and beautiful mathematical framework - a new kind of logic that we could try to apply to the physical world.” Schilling adds, “Isham’s hope is that the new approach will lead to a reformulation of quantum theory, which would pave the way for a decent theory of quantum gravity, without ugly infinities.”
However, Nobel Prize winning physicist Gerard ’t Hooft of Utrecht University in the Netherlands insists that yet another mathematical language is irrelevant. Consequently, he thinks that Isham’s new formalism isn’t likely to be very useful. Schilling quotes ‘t Hooft’s expression of skepticism:
Our biggest problem is how to formulate our questions,” he says. “What exactly do you want to know, and which questions are you able to answer? Isham believes another mathematical language may help, but I don’t think so. It sounds a bit as if describing the world in German is better than in Chinese.
Others, like Robbert Dijkgraaf of the University of Amsterdam, are likewise skeptical. Schilling quotes his opinion of Isham’s approach as a “long shot,” even though it “provides a fundamental way of describing quantization.” Woit too expresses doubt. He writes:
I have to say that, like ‘t Hooft and Dijkgraaf who are quoted in the article, I’m skeptical about this kind of thing, since topos theory is such a general formalism that I don’t see how it is going to provide the sort of non-trivial new idea that people are looking for.
Hence, it seems clear to skeptics, at this point in time, that a new mathematical formalism, or new language, for physics, no matter how good it might be, cannot overcome the limitations now facing physicists.
But perhaps the reason why a new formalism is unlikely to help much is especially easy to understand, when we realize that the discrete/continuous duality of nature is not only reflected in the magnitudes of nature, but also in algebra itself. Ironically enough, something that is not generally recognized today is that Hamilton’s contributions to the modern mathematical formalisms of mathematics and physics, as fundamental and indispensable as they are, were nevertheless viewed by him, at least early on in his career, as almost repugnant expedients, as long as no intuitive science of numbers, or algebra, exists.
The powerful genius of this man was troubled by the fact that, while geometry is firmly grounded in the science of space, algebra has no similar foundation, but contains inconsistencies and even contradictions. In the introductory section of his remarkable paper, “Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time,” Hamilton describes his complaint, by first making the clear distinction between formalism, which he refers to as philologicism, and theory, which he regards as contemplated truth:
The Study of Algebra may be pursued in three very different schools, the Practical, the Philological, or the Theoretical, according as Algebra itself is accounted an Instrument, or a Language, or a Contemplation; according as ease of operation, or symmetry of expression, or clearness of thought, (the agere, the fari, or the sapere,) is eminently prized and sought for. The Practical person seeks a Rule which he may apply, the Philological person seeks a Formula which he may write, the Theoretical person seeks a Theorem on which he may meditate.
The felt imperfections of Algebra are of three answering kinds. The Practical Algebraist complains of imperfection when he finds his Instrument limited in power; when a rule, which he could happily apply to many cases, can be hardly or not at all applied by him to some new case; when it fails to enable him to do or to discover something else, in some other Art, or in some other Science, to which Algebra with him was but subordinate, and for the sake of which and not for its own sake, he studied Algebra.
The Philological Algebraist complains of imperfection, when his Language presents him with an Anomaly; when he finds an Exception disturb the simplicity of his Notation, or the symmetrical structure of his Syntax; when a Formula must be written with precaution, and a Symbolism is not universal.
The Theoretical Algebraist complains of imperfection, when the clearness of his Contemplation is obscured when the Reasonings of his Science seem anywhere to oppose each other, or become in any part too complex or too little valid for his belief to rest firmly upon them; or when, though trial may have taught him that a rule is useful, or that a formula gives true results, he cannot prove that rule, nor understand that formula: when he cannot rise to intuition from induction, or cannot look beyond the signs to the things signified.
If we did not know better, we would think he was referring to the theory of quantum mechanics found in modern theoretical physics, not the theory of algebra. However, the same perplexing situation existed then in algebra that exists today in physics; that is, the formula of quantum physics, the wave equation, gives true results, but, in reality, physicists have no clue as to what the foundation of quantum physics might be. In Hamilton’s view, the reasonings of algebraic science were just as perplexing. The rules for adding, subtracting, multiplying and dividing worked, but the mathematician couldn’t “rise to intuition from induction,” by “looking beyond the signs to the things signified,” any more than the modern physicist can understand nature today, by studying the laws of quantum mechanics. In his paper, Hamilton clearly describes the problems of algebra he seeks to address:
These remarks have been premised, that the reader may more easily and distinctly perceive what the design of the following communication is, and what the Author hopes or at least desires to accomplish. That design is Theoretical, in the sense already explained, as distinguished from what is Practical on the one hand, and what is Philological upon the other.
The thing aimed at, is to improve the Science, not the Art nor the Language of Algebra. The imperfections sought to be removed, are confusions of thought, and obscurities or errors of reasoning; not difficulties of application of an instrument, nor failures of symmetry in expression. And that confusions of thought, and errors of reasoning, still darken the beginnings of Algebra, is the earnest and just complaint of sober and thoughtful men, who in a spirit of love and honour have studied Algebraic Science, admiring, extending, and applying what has been already brought to light, and feeling all the beauty and consistence of many a remote deduction, from principles which yet remain obscure, and doubtful.
For it has not fared with the principles of Algebra as with the principles of Geometry. No candid and intelligent person can doubt the truth of the chief properties of Parallel Lines, as set forth by Euclid in his Elements, two thousand years ago; though he may well desire to see them treated in a clearer and better method. The doctrine involves no obscurity nor confusion of thought, and leaves in the mind no reasonable ground for doubt, although ingenuity may usefully be exercised in improving the plan of the argument.But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and Imaginaries, when set forth (as it has commonly been) with principles like these: that a greater magnitude may be subtracted from a less, and that the remainder is less than nothing; that two negative numbers, or numbers denoting magnitudes each less than nothing, may be multiplied the one by the other, and that the product will be a positive number, or a number denoting a magnitude greater than nothing; and that although the square of a number, or the product obtained by multiplying that number by itself, is therefore always positive, whether the number be positive or negative, yet that numbers, called imaginary, can be found or conceived or determined, and operated on by all the rules of positive and negative numbers, as if they were subject to those rules, although they have negative squares, and must therefore be supposed to be themselves neither positive nor negative, nor yet null numbers, so that the magnitudes which they are supposed to denote can neither be greater than nothing, nor less than nothing, nor even equal to nothing.
It must be hard to found a Science on such grounds as these, though the forms of logic may build up from them a symmetrical system of expressions, and a practical art may be learned of rightly applying useful rules which seem to depend upon them.
So, then, what’s the answer? How do we “found a Science” of algebra that will enable us to “rise to intuition from induction?” As it turns out, Hamilton’s answer to this question is the same as Larson’s answer to the similar question, asked in relation to physics: The answer is to be found in the recognition that numbers correspond to the order of progression, as well as to the magnitudes of isolated quantities. Only on this basis, Hamilton was convinced, could algebra rise to the level of science, comparable to the science of geometry. Yet, whether or not this conclusion is justified, the argument that, without a more intuitive basis, algebra causes reason to stare, is powerful. He writes:
So useful are those rules, so symmetrical those expressions, and yet so unsatisfactory those principles from which they are supposed to be derived, that a growing tendency may be perceived to the rejection of that view which regarded Algebra as a Science, in some sense analogous to Geometry, and to the adoption of one or other of those two different views,which regard Algebra as an Art, or as a Language: as a System of Rules, or else as a System of Expressions, but not as a System of Truths, or Results having any other validity than what they may derive from their practical usefulness, or their logical or philological coherence.
Opinions thus are tending to substitute for the Theoretical question,—“Is a Theorem of Algebra true?” the Practical question,—“Can it be applied as an Instrument, to do or to discover something else, in some research which is not Algebraical?” or else the Philological question,—“Does its expression harmonise, according to the Laws of Language, with other Algebraical expressions?”
Yet a natural regret might be felt, if such were the destiny of Algebra; if a study, which is continually engaging mathematicians more and more, and has almost superseded the Study of Geometrical Science, were found at last to be not, in any strict or proper sense, the Study of a Science at all: and if, in thus exchanging the ancient for the modern Mathesis, there were a gain only of Skill or Elegance, at the expense of Contemplation and Intuition.Indulgence, therefore, may be hoped for, by any one who would inquire, whether existing Algebra, in the state to which it has been already unfolded by the masters of its rules and of its language, offers indeed no rudiment which may encourage a hope of developing a Science of Algebra: a Science properly so called: strict, pure and independent; deduced by valid reasonings from its own intuitive principles; and thus not less an object of priori contemplation than Geometry, nor less distinct, in its own essence, from the Rules which it may teach or use, and from the Signs by which it may express its meaning.
Hamilton found that the “rudiment” of the less than inspiring existing algebra, “which may encourage a hope of developing a science of algebra,” is “the intuition of time.” He explains how he arrived at this conclusion:
This belief involves the three following as components: First, that the notion of Time is connected with existing Algebra; Second, that this notion or intuition of Time may be unfolded into an independent Pure Science; and Third, that the Science of Pure Time, thus unfolded, is co-extensive and identical with Algebra, so far as Algebra itself is a Science. The first component judgment is the result of an induction; the second of a deduction; the third is the joint result of the deductive and inductive processes.
Hamilton argues the first point, that time is connected with existing algebra, by observing the nature of calculus, which has to do with change over time, a continuous progression:
The argument for the conclusion that the notion of Time is connected with existing Algebra, is an induction of the following kind. The History of Algebraic Science shows that the most remarkable discoveries in it have been made, either expressly through the medium of that notion of Time, or through the closely connected (and in some sort coincident) notion of Continuous Progression. It is the genius of Algebra to consider what it reasons on as flowing, as it was the genius of Geometry to consider what it reasoned on as fixed.
Euclid defined a tangent to a circle, Apollonius conceived a tangent to an ellipse, as an indefinite straight line which had only one point in common with the curve; they looked upon the line and curve not as nascent or growing, but as already constructed and existing in space; they studied them as formed and fixed, they compared the one with the other, and the proved exclusion of any second common point was to them the essential property, the constitutive character of the tangent.
The Newtonian Method of Tangents rests on another principle; it regards the curve and line not as already formed and fixed, but rather as nascent, or in process of generation: and employs, as its primary conception, the thought of a flowing point. And, generally, the revolution which Newton made in the higher parts of both pure and applied Algebra, was founded mainly on the notion of fluxion, which involves the notion of Time. Before the age of Newton, another great revolution, in Algebra as well as in Arithmetic, had been made by the invention of Logarithms; and the “Canon Mirificus” attests that Napier deduced that invention, not (as it is commonly said) from the arithmetical properties of powers of numbers, but from the contemplation of a Continuous Progression; in describing which, he speaks expressly of Fluxions, Velocities and Times.
In a more modern age, Lagrange, in the Philological spirit, sought to reduce the Theory of Fluxions to a system of operations upon symbols, analogous to the earliest symbolic operations of Algebra, and professed to reject the notion of time as foreign to such a system; yet admitted that fluxions might be considered only as the velocities with which magnitudes vary, and that in so considering them, abstraction might be made of every mechanical idea. And in one of his own most important researches in pure Algebra, (the investigation of limits between which the sum of any number of terms in Taylor’s Series is comprised,) Lagrange employs the conception of continuous progression to show that a certain variable quantity may be made as small as can be desired.
And not to dwell on the beautiful discoveries made by the same great mathematician, in the theory of singular primitives of equations, and in the algebraical dynamics of the heavens, through an extension of the conception of variability, (that is, in fact, of flowingness,) to quantities which had before been viewed as fixed or constant, it may suffice for the present to observe that Lagrange considered Algebra to be the Science of Functions, and that it is not easy to conceive a clearer or juster idea of a Function in this Science, than by regarding its essence as consisting in a Law connecting Change with Change.
But where Change and Progression are, there is Time. The notion of Time is, therefore, inductively found to be connected with existing Algebra.
That algebra, connected with time, can be “unfolded in an independent pure science,” is a logical deduction from the intuition that only three possibilities exist in relation to different moments of time: One moment may be identified as the same, earlier than, or later than another. Hamilton explains:
The argument for the conclusion that the notion of time may be unfolded into an independent Pure Science, or that a Science of Pure Time is possible, rests chiefly on the existence of certain a priori intuitions, connected with that notion of time, and fitted to become the sources of a pure Science; and on the actual deduction of such a Science from those principles, which the author conceives that he has begun.
Whether he has at all succeeded in actually effecting this deduction, will be judged after the Essay has been read; but that such a deduction is possible, may be concluded in an easier way, by an appeal to those intuitions to which allusion has been made. That a moment of time respecting which we inquire, as compared with a moment which we know, must either coincide with or precede or follow it, is an intuitive truth, as certain, as clear, and as unempirical as this, that no two straight lines can comprehend an area.
The notion or intuition of Order in Time is not less but more deep-seated in the human mind, than the notion of intuition of Order in Space; and a mathematical Science may be founded on the former, as pure and as demonstrative as the science founded on the latter. There is something mysterious and transcendent involved in the idea of Time; but there is also something definite and clear: and while Metaphysicians meditate on the one, Mathematicians may reason from the other
In our own work, following Larson, we have developed a new kind of algebra at the LRC, based on continuous progression and the operational interpretation of number, but only recently did we discover that no less a figure than Hamilton had investigated and justified the mathematics of progression. Of course, there are major differences, but the idea of displacements, the primary concept of the mathematical science of time, as Hamilton called it, is the central idea of both. Hamilton’s conclusion was that the mathematical science of time is coextensive and identical with the mathematical science of algebra; that is, the mathematical science of order in progression is just that, a science, not a formalism. He writes:
That the Mathematical Science of Time, when sufficiently unfolded, and distinguished on the one hand from all actual Outward Chronology (of collections of recorded events and phenomenal marks and measures), and on the other hand from all Dynamical Science (or reasonings and results from the notion of cause and effect), will ultimately be found to be co-extensive and identical with Algebra, so far as Algebra itself is a Science: is a conclusion to which the author has been led by all his attempts, whether to analyse what is Scientific in Algebra, or to construct a Science of Pure Time.
It is a joint result of the inductive and deductive processes, and the grounds on which it rests could not be stated in a few general remarks. The author hopes to explain them more fully in a future paper; meanwhile he refers to the present one, as removing (in his opinion) the difficulties of the usual theory of Negative and Imaginary Quantities, or rather substituting a new Theory of Contrapositives and Couples, which he considers free from those old difficulties, and which is deduced from the Intuition or Original Mental Form of Time: the opposition of the (so-called) Negatives and Positives being referred by him, not to the opposition of the operations of increasing and diminishing a magnitude, but to the simpler and more extensive contrast between the relations of Before and After, or between the directions of Forward and Backward; and Pairs of Moments being used to suggest a Theory of Conjugate Functions, which gives reality and meaning to conceptions that were before Imaginary, Impossible, or Contradictory, because Mathematicians had derived them from that bounded notion of Magnitude, instead of the original and comprehensive thought of Order in Progression.
What Hamilton discovered, that the opposition of negatives and positives may be referred to relations of before and after, instead of “the operations of increasing and diminishing a magnitude,” and that “pairs of moments” suggest a “theory” that “gives reality and meaning to conceptions that were before imaginary, impossible, or contradictory,” leads to a new understanding, which is that the bounded notion of magnitude, an isolated quantity, is limited and misleading compared to the “original and comprehensive thought of order in progression.”
Nevertheless, as far as I can tell, this idea, so vital to Hamilton, has lain dormant ever since the day he discovered it, even though it clearly may be, to put it in the words of Woit above, “the sort of non-trivial new idea that everybody is looking for.”
The prospect that the new idea of a mathematical science of progression opens the door to understanding the mystery of nature’s discrete/continuous duality was not apparent until Larson’s work revealed the true nature of space: It is simply the reciprocal of time. Given the power of understanding that the mathematical science of time is coextensive to, and identified with, the science of algebra, as discovered by Hamilton intuitively, prepares us in a remarkable way to understand the reciprocal relation of two such progressions, which consequently forms the basis for the structure of the physical universe, a structure consisting of nothing but motion.
At the same time, it gives us a renewed appreciation for the value of intuition in science. One dimension of that insight that we probably should always keep in mind is that many times great things are brought to pass by very small means.
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